the second fundamental form
When a surface sits inside a bigger space, there are two kinds of bending. The first fundamental form (the induced metric) measures distances WITHIN the surface — its intrinsic geometry. The second fundamental form measures how the surface curves AWAY from being flat inside the ambient space — its extrinsic geometry. It is the precise answer to 'how does this submanifold bend within the space that contains it?'
Let M be a submanifold of a Riemannian manifold (N, g-bar) with the induced metric, and let nabla-bar and nabla be the Levi-Civita connections of N and M. For tangent vector fields X, Y on M, the second fundamental form is the normal-component remainder II(X, Y) = nabla-bar_X Y - nabla_X Y, a symmetric vector-valued bilinear form taking values in the normal bundle. In words: differentiate Y along X using the ambient connection, then subtract off the part tangent to M; what is left, pointing out of M, is II(X,Y). For a hypersurface with unit normal nu, one often writes the scalar second fundamental form h(X,Y) = g-bar(II(X,Y), nu).
Why it matters: II is the bridge between intrinsic and extrinsic geometry. Its trace (with respect to the metric) is the mean curvature, governing minimal surfaces and soap films; its eigenvalues are the principal curvatures. Crucially, the Gauss equation expresses the INTRINSIC curvature of M in terms of the ambient curvature and II, so a surface's intrinsic bending is partly explained by how it sits in space. Honesty: II is extrinsic — it depends on the embedding, not just on M's own metric. Two isometric surfaces (a flat sheet and the same sheet rolled into a cylinder) have the SAME intrinsic geometry but DIFFERENT second fundamental forms; that the intrinsic Gaussian curvature comes out equal anyway is the content of Gauss's theorema egregium.
Roll a flat sheet of paper into a cylinder. Intrinsically nothing changed — the induced metric is still flat, Gaussian curvature 0, and you cannot stretch or tear. But the second fundamental form changed completely: the flat sheet has II = 0, while the cylinder has a nonzero principal curvature 1/r in the wrapping direction and 0 along the axis. That is extrinsic bending with zero intrinsic curvature.
Cylinder vs flat sheet: same intrinsic metric (Gaussian curvature 0), different second fundamental form.
The second fundamental form is EXTRINSIC: it depends on the embedding, not just the metric of M. Isometric submanifolds can have different II — the surprise of theorema egregium is that their intrinsic curvature still agrees.